Math is many things: beautiful, fundamental, universal, but ultimately, math is hard. So hard in fact that mathematicians often need to phone up their other mathematician friends for help. The idea of the crazy-maned recluse furiously working in his office alone is an outdated one. While some modern mathematicians still fit this bill, collaboration is increasingly the norm. By the year 2000, the number of mathematics papers with a single author had shrunk to 50%. More and more people are tackling difficult math problems as a team.
No one embodies the idea of mathematical collaboration more than Paul Erdös (pronounced err-desh or air-dish), a Hungarian mathematician who lived in the twentieth century. A legendary figure in mathematics, Erdös published around 1500 papers and had around 500 co-authors. To contrast, most mathematicians write 7 papers in their entire life! Erdös was heavily in support of working together to solve math problems and questions, and also had incredible mathematical taste. He asked very interesting questions and would often attach a dollar amount to the questions. If you were clever enough to solve one of these Erdös questions, Paul Erdös himself would send you a cheque. These cheques were so revered in mathematics that often people frame them rather than cash them. More information on his very interesting life is available here.
It is in this context that the idea of the Erdös number emerged. In the 60s, one of Erdös’ friends presented the idea of the Erdös number to measure how close you were to Erdös in terms of collaboration. So Erdös himself has Erdös number 0 and all of his 511 direct co-authors have Erdös number 1. Now if you co-author a paper with say, Terrence Tao who has Erdös number 1, you would have Erdös number 2! If, like me, you have never published a paper, or you have never co-authored a paper, you would have an infinite Erdös number.
I would say that most mathematicians know their Erdös number (or can easily find it out); it is kind of like a feather in their hat. Your Erdös number on its own won’t get you a job, a diploma or a date, but it does look nice. Some people have been known to write papers with collaborators simply to lower their Erdös number (I think someone was selling the opportunity to co-author with them on Ebay) but this is relatively rare.
Now for some facts (all of which I found on the helpful Erdös Number Project). The smallest Erdös number is of course 0, but the largest is 13 (or 15 depending on what you consider to be a paper). Arguably the most famous mathematician, Andrew Wiles who solved Fermat’s Last Theorem, has Erdös number 3. Looking at the who’s who of brilliant mathematicians we see that every Fields medalist (kind of the math Nobel prize) has had Erdös number 5 or less, with most having number 2 or 3. In contrast the upper bound for the Nobel Prize in Medicine is 11.
Many non-mathematicians have Erdös numbers as well. Bill Gates, Stephen Hawking and Carl Sagan each have number 4. Some fields are even known for having low Erdös numbers. Biology and Linguistics are examples, as in Linguistics, Noam Chomsky has Erdös number 4.
Even more surprisingly, there are many actors with Erdös numbers. Natalie Portman (5) and Colin Firth (6) are counted among the various co-co-co-co-collaborators of Erdös. This leads to the inevitable connection with the “6 degrees of Kevin Bacon” game. You remember the game where I give you an actor and you try to connect them to Kevin Bacon by saying “A was in a film with B who was in a film with C … who was in a film with Kevin Bacon”.
Thus you have an actor’s “Bacon number”, the length of the shortest path of that sort to Kevin Bacon. Going even further you have “Erdös-Bacon numbers” (it’s a real thing, look it up). This is the sum of a person’s Erdös number and their Bacon number. Of the people I mentioned, Carl Sagan has EB number 7, Stephen Hawking has EB number 7 (if you count his appearance on the Simpsons), Colin Firth has EB number 7 and Natalie Portman has EB number 6. The smallest (non-disputed) EB number is 3 which belongs to Bruce Reznick a math professor with Erdös number 1 that appeared as an extra in Pretty Maids all in a Row, which gives him Bacon number 2.
By the way, this is from the Wikipedia plot synopsis of Pretty Maids All in a Row: “The story is set in Oceanfront High School, a fictitious American high school in the height of the sexual revolution. Young female students are being targeted by an unknown serial killer. Meanwhile, a male student called Ponce is experiencing sexual frustration, surrounded by a seemingly unending stream of beautiful and sexually provocative classmates.”
To end, let’s examine the famous baseball player Hank Aaron (who in my mind is still the all time leader in home runs) who has Bacon number 2 (he was in “Summer Catch” with Susan Gardner who was in “In the Cut” with Kevin Bacon). Hank Aaron has also signed the same baseball as Paul Erdös, giving him a tenuous Erdös number 1. So you could say that Hank Aaron also has an Erdös-Bacon number of 3!